Cmjv02i01p0145

J. Comp. & Math. Sci. Vol.2 (1), 145-152 (2011)

Hankel Convolution on Ultradistribution Spaces with Exponential Growth S. K. UPADHYAY Department of Applied Mathematics, Institute of Technology, CIMS (DST), Banaras Hindu University Varanasi – 221 005 (India) ABSTRACT µ

µ

In this paper the properties of the spaces of type H ω and Q ω are obtained by using the theories of Hankel convolution and Hankel transformation.

Key Words: Ultradistribution, Hankel transformation, Hankel Convolution, Bessel functions.

AMS. Classification: 46F12.

1. INTRODUCTION A characterization of Hankel convolution on the spaces of distributions with exponential growth was given by J. J. Betancor and Mesha1. This characterization naturally yields a characterization for ultradistribution spaces of type H µω and Qµω .

The theory of ultradistributions have been investigated by Beurling2, Björck3 and Roumieu7. These ultradistributions are generalizations of Schwartz distributions. A unification of Beurling-Björck theory and Roumieu theory was done by Komatsu4. The Hankel transformation of ultradistributions was introduced by Pathak and Pandey6. The spaces of type Xµ and Qµ were defined by Betancor-Mesha1, and studied distributional Hankel convolution and Hankel transformation on these spaces. The main objective of this paper is to study the

Beurling type ultradistribution spaces H µω and Qµω and their Hankel transformation and Hankel convolution transformation on these spaces.

Now, we give the definition of Hankel convolution from5. Let ∞ −µ−

Dµ (x, y, z) = ∫ t 0

1 2

1

1

(xt) 2 J µ (xt)(yt) 2

1 2

J µ (yt) (zt) J µ (zt) dt

(1.1)

provided that the above integral exists. The translation f(x,y) of the function f is defined by ∞

f (x, y) = ∫ f (z)Dµ (x, y, z) dz ∀ x, y ∈ I , (1.2) 0

and Hankel convolution f # φ of f and φ is the function ∞

(f # φ)(x) = ∫ f (x, y)φ(y) dy .

0

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)

(1.3)

S. K. Upadhyay, J. Comp. & Math. Sci. Vol.2 (1), 145-152 (2011)

146

Next, we recall the definition of spaces of type Xµ and Qµ. The space Xµ which is the set of all smooth-complex valued function φ (x), x ∈ I such that k

γ

µ m,k

1

d  −µ− 2  (φ) = sup exp(kx)  x −1 φ(x)  x dx  x∈I 

< ∞, ∀m, k ∈ N 0 (1.4) From [1, p. 37] we can say that Xµ is Frechet space and nuclear. The semi-norm of above space is given as

ηµm,k (φ) = sup exp(kx)x x∈I

1 −µ− 2

Sµm φ(x)

, (1.5)

< ∞, ∀φ ∈ X µ and k, m ∈ N 0 where Sµ denotes the Bessel operator induce on Xµ the same topology is defined by

{γ }

µ m,k k ,m∈N 0

.

In this paper a test function space Q generalizing the space Qµ is defined. It is shown that the Hankel transformation hµ is µ µ an isomorphism from H ω and Q ω . Several properties of Hankel convolution transform and generalized Hankel transform of µ µ ultradistribution in H ω space and (H ω ) ' space are discussed. µ ω

2. PROPERTIES OF HANKEL µ µ TRANSFORMATION IN H ω and Q ω SPACES In this section we give some µ µ properties of H ω and Q ω spaces by using the theory of Hankel transformation.

H

µ ω

valued function defined on I = (0, ∞) possessing the following properties: 0 ≤ ω(ξ + η) ≤ ω(ξ) + ω(η) ∀ ξ, η∈ I (2.1)

From5 we recall the definition of space. Let ω be a continuous real

∫

∞

0

ω(ξ) dξ < ∞ 1 + ξ2

(2.2)

ω(ξ) > a + b log (1 + ξ) for some real a, b > 0.

(2.3) µ

Now, the space H ω denotes the set

of all C∞-complex valued function φ on I such that the following norm is satisfied. k

d  −µ− 1  γ µλ,k (φ) = sup exp[λω(x)]  x −1  x 2 φ(x) < ∞ dx  x∈I 

(2.4) for all non-negative real number λ and nonnegative integers k. µ The space Qω which is the set of all even entire analytic function hµ φ such that ρmk,λ (φ) = sup | exp λω(| z |) z |Im(z)| ≤ k

−µ−

1 2

(h µ φ)(z) | < ∞

(2.5) for all m, k ∈ N0 and ω (|z|) is satisfied the same properties from 2.1 and 2.3.

Theorem 2.1. The Hankel transformation µ µ hµ is an isomorphism from H ω onto Qω for

1 µ≥− .

2 To prove the above theorem, following Lemmas will be obtained.

Lemma 2.2. The Hankel transformation hµ is a continuous linear mapping from µ µ H ω int o Qω .

Proof. Let φ∈Hµω . Then

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)

S. K. Upadhyay, J. Comp. & Math. Sci. Vol.2 (1), 145-152 (2011)

exp[ λω (| z |)z

−µ−

1 2

( λε ) z m! m =0 ∞

m

exp[ λ C( ε )] ∑

(λε)  ≤ exp[λC(ε)] ( ∑ m! m =0

 −1 d  x  x dx  

1 −µ− 2

m

 

  x −1

m

d   x dx 

1 −µ− 2

∫

 x

2 µ+ m +1

0

(h µ φ )(z)

∞

−µ

J µ (zx) x

0

m =0

(λε) m exp (− − aN b ) m!

γ µλ+ I + N b (φ) Cω ) ∞

e − kx | (zx)−µ J µ (zx) | ekx x µ+ 2m +1



≤ exp[λC(ε)]( ∑ Cµ,k,m m =0

γ

µ λ+1+ N b

(λε) m exp (− − aN b ) m!

(φ))

Now, choosing

∞ (λε)m kn A ∑ µ ,m ∑ m = 0 m! n = 0 n! ∞

d  −µ− 1  x  x −1  x 2 φ(x) dx). dx   n

−µ−

1 2

(h µ φ)(z) ≤ exp[λC(ε)]

∞ (λε) m kn (∑ ( sup | (zx)−µ J µ (zx) | (∑ ) m! Im(z) ≤ k m=0 n =0 n ! ∞

∞

−1 N ∫ x exp[λω(x)](x 0

1

d m −µ− 2 ) x φ(x) dx) dx

ε < (λCµ ,k,m,ω γ µλ+1+ N b ,m (φ)) m , m ≥ 1 we have

sup | exp[λω(| z |) z

|Im(z)|≤ k

∞

(λε) m A µ ,me k ∫ exp(−aN / b)| exp[(λ + 1 + N / b)ω(x)] m = 0 m!

0 1

d m −µ− 2 ) x φ(x) | exp (−ω(x)) dx) dx ∞ (λε)m  ≤ exp[λC(ε)]( ∑ Cµ ,k,m exp (− aN b ) m! m=0

−µ−

1 2

(h µ φ)(z) |

∞

1 ≤ C exp[λC(ε)]( ∑ exp(− aN b )) < ∞ m = 0 m!

.

Lemma 2.2.

The inverse Hankel −1 transformation h µ is continuous linear µ

µ

mapping from Qω into H ω .

Proof. Let φ∈Hµω . Then [1, pp. 38-39] gives m 1   −1 d   −µ− 2 x x φ(x))     dx     (2.6) ≤ C e−ηx

≤ exp[λC(ε)]

(x −1



0

|Im(z)|≤ k

∞

∞

φ(x) |( ∫ exp(−ω(x))dx)

≤ exp[λC(ε)]( ∑ Cµ,k,m

m + 2 µ+1

φ(x) dx)

sup exp[λω(| z |)]z

(∑

1 2

∞

Now, assume that N > 2µ + m + 1 is an integer such that



x

−µ−

d m ) dx

1

 ≤ exp[λC(ε)]( ∞

∫ (zx)

x∈I



φ(x) dx)

 ≤ exp[λC(ε)]( ∑ (λε) ∫0 m = 0 m! ∞

1 m −µ− 2

m ∞

∞

m

sup| exp[(λ + 1 + N b )ω(x)](x −1

(h µ φ )(z) ≤

147

∫

∞

−∞

(ξ + iη)m (h µ φ)(ξ + iη) dξ

Applying inequality ω(x) ≤ εx + C(ε) , we get m 1   −1 d   −µ− 2 x x φ(x))     dx     ≤ C exp ( ( k + 1) C ( ε ) ) .exp ( − ( k + 1) ω(x) )

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)

S. K. Upadhyay, J. Comp. & Math. Sci. Vol.2 (1), 145-152 (2011)

148

(∫

+∞

−∞

| ( ξ + i ( k + 1) )

m

( h φ) ( ξ + i ( k + 1) ) | dξ ) . µ

Thus, d m -µ - 1 2 ) x φ(x) | dx ≤ C.Cε exp ( −ω(x) )

| exp ( kω(x) ) (x -1

(∫

+∞

−∞

| ( ξ + i(k + 1)

( ξ + i(k + 1) ≤ Dε ξ

∫

−∞

µ

sup exp[λω(| t |)]t −µ− 2 φx (t) ( h µ φ ) (t) 1

m +µ+ 1 2

|Im(t )|≤ k

( h φ ) ( ξ + i ( k + 1) ) | dξ) µ

|Im(z)|≤ k+1

( h φ ) ( ξ + i ( k + 1) ) | |Im(z)|≤ k+1

2

( h φ ) ( ξ + i ( k + 1) ) | . µ

Hence,

d m −µ− 12 ) x φ(x) | dx µ ≤ Dω,ε ρ(k +1),( λ+1) (φ).

| exp[kω(x)](x −1

(2.7)

≤ sup | exp[(λ + q)ω(| t |)]t −µ−

1

2

|Im( t )|≤ k

≤ Cq sup | exp[(λ + q)ω(| t |)]t µ−

( h φ ) (t) | µ

1

2

x ∈ (0,1) and k, m ∈ (0,1) we obtain d m −µ− 12 ) x φ(x) | dx

( h φ ) (t) | µ

This shows that

φx (t) ( h µ φ ) (t) ∈ Qµω .

Theorem 2.4. For every x ∈ I = (0, ∞ ) the Hankel translation τx is continuous linear µ mapping from H ω into itself.

Proof. Let φ∈ Hµω . Then using the arguments of [1,p.40] the Hankel translation is defined as

So that for every | exp[kω(x)](x −1

|Im( t )|≤ k

< ∞.

µ

≤ Dε .Dω sup | exp[(λ + 1)ω(| ξ + i(k + 1) |)] 1

≤ sup | exp[−qω(| t |)]φ x (t) |

|Im( t )|≤ k

exp ( −ω(| ξ + i(k + 1) |))dξ )

ξ −µ−

and ( h µ φ) (t) ∈ Qµω . Then φx (t) ( h µ φ) (t) ∈ Qµω .

Proof. Since φ(t) ∈ Qω . Therefore we have

sup | exp[(λ + 1)ω(| ξ + i(k + 1) |)]

−µ− 1 2

+∞

−µ− 1 2

)

| exp ( −qω(| t |))φx (t) |) ≤ Cq , ∀t ∈ C, q>0,

(2.8)

≤ Dω,ε ρ1,µ λ+1 (φ)

( τx φ )

(y) = h µ [t −µ−

1

2

( xt )

1

2

J µ (xt) (h µ φ)(t)]

µ

Since φ∈ H ω . Therefore Theorem 2.1 says

(

)

that h µ φ ∈ Qµω . Using Lemma 2.3 we get

From Lemma 2.1 and 2.2, we find that hµ = h µ−1 and H µω is a subspace of Hµ . This

t

implies that the Hankel transformation hµ is µ µ an isomorphism from Qω onto H ω .

Hence, by the property of Hankel transformation and Theorem 2.1, we obtain

−µ

Lemma 2.3. Let φx (t) = (xt) J µ (xt), t ∈ C be an entire analytic function of t which satisfy the inequality

−µ− 1

(

2

( xt )

1

2

J µ (xt) (h µ φ)(t) ∈ Qµω .

)

h µ t −µ− 2 (xt) 2 J µ (xt) ( h µ φ ) (t) ∈ Hµω .

1

1

Theorem 2.5

f ∈ H µω and g ∈ Hµω then f#g ∈ H µω .

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)

S. K. Upadhyay, J. Comp. & Math. Sci. Vol.2 (1), 145-152 (2011)

Proof. Let ( h µ f ) (t) ∈ Qµω and ( h µ g ) (t) ∈ Qµω .

Then (2.5) gives 1 ρ ( f # g ) = sup exp[λω(| t |)t µ− 2 h µ (f # g) (t)

149

µ

Let φ∈ H ω . Then the Hankel convolution of T and φ can be written in the following form:

( T # φ ) (x) =< T, τx φ >

|Im(t ) ≤ k

= sup exp[λω(| t |)t

−2 µ−1

|Im( t ) ≤ k

(h µ f )(t)(h µ g) (t)

≤ sup exp[(λ − r)ω(| t |)t

−µ− 1

2

|Im(t )≤ k

(h µ f )(t)

× sup exp[rω(| t |)]t −µ− 2 (h µ g)(t)

∞

=

∫

=

∫

0 ∞ 0

T ( y ) ( τ x φ ) ( y )dy S µk , y ( e x p [ λ ω ( y ) ] y - µ - 2 f k 1

( y )) (τ x φ ) ( y ) d y

1

|Im(t ) ≤ k

= ( − 1) k

∫

∞ 0

e x p [ λ ω (y)]y - µ - 2 f k 1

( y )S µk ( τ x φ ) ( y ) d y

= ρ ( h µ f ) .ρ ( h µ g ) < ∞.

From [1, pp. 41-42] we have Therefore,

h µ ( f # g ) ∈ Qµω .

∞

( T # φ ) ( x ) = ( − 1) k ∫0

Theorem 2.1 shows that

Theorem 2.6 Let 1 T ∈ ( H µω ) ' and φ ∈ Hµω then x -µ - 2 H

1

2

h µ [(xt ) − µ J µ ( xt)(h µ φ )(t )]( y)dy.

f # g ∈ Hµω .

( T # φ ) ∈ θω

exp[ λ ω ( y)]y − µ −

where θωH

So that

| ( x −1

d n − µ − 12 ) x ( T # φ )(x )| dx

≤ sup | exp[(λ + 1)ω(y)]y −µ− 2 h µ [(xt) −µ− n 1

denotes the multiplier.

y∈I

( )

∞

Proof. Let T ∈ H µω ' then k

(

T = ∑ Sµk exp[λω(x)]x r =0

J µ (xt)t 2n + 2k (h µ φ)(t)](y) | ∫ exp[ −ω(y)]dy 0

−µ− 1

2

fk

)

where fk is bounded on I for 0 ≤ k ≤ r.

Hence, we prove the following result

(

T = S (exp[λω(x)]x k µ

−µ− 1 2

fk )

)

where fk is bounded on I and λ, k ∈ N0.

≤ C s u p | e x p [( λ + 1) ω (s )]s − µ − 2 h µ 1

y∈ I

[( x t ) − µ − n J µ + n ( x t ) t 2 n + 2 k ( h µ φ )( t )](s ) | for s = y + iv.

By definition (2.5) the right hand side for the above expression gives

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)

S. K. Upadhyay, J. Comp. & Math. Sci. Vol.2 (1), 145-152 (2011)

150

d n − µ − 12 ) x (T # φ )(x ) |≤ C ρ µl ,m dx [h µ (t 2 n + 2 k (xt ) − µ − n J µ + n (xt)(h µ φ )( t))] | (x − 1

Next, we introduce distributional space

( H ) ' which is a subspace of ( H ) '

≤ C ρ µl ,m (h µ φ ) exp(x l ) ≤ C µ ,l

∞

µ ,* ω,m

x plp p!

∑ p= 0

Proof. The proof of above theorem can be derived by using the technique of [1, pp. 4245]. µ ω

such that

S # φ ∈ H µω for S ∈ ( H µω,*,m ) ' and φ ∈ H µω .

(

≤ C µ ,l e l .exp[ λ ω (x)]

)

Now, the space H µω,*,m ' consists of

≤ D µ , l e x p [ λ ω (x )].

all complex-valued smooth function φ (x), x ∈ I such that

Hence,

α km,µ (φ) = exp[mω(x)]x −µ− 2 Sµk φ(x) |< ∞ 1

d n − µ − 12 ) x dx

| ex p [ − λ ω ( x )]( x )] (x -1 (T # φ )( x )(x )| ≤ D µ , l .

1

2

H

.

( ) ' and is defined by

to an element of H

µ ω

∞

<T#φ,ψ >= ∫ (T#φ)(x) ψ(x)dx, ψ ∈ H µω .

0

(

)

Theorem2.7. If T ∈ H µω ' and φ ∈ H µω µ ω

< T # φ ,ψ > = < T ,φ # ψ > , ψ ∈ H ,

( T # φ) = x

h (T).h µ φ ' µ

T ∈ ( H µω ) ' the mapping φ → T#φ is from

( H ) into ( H ) ' for µ ω

considering strong topology.

. Like Hµω space

( H ) is µ ,* ω ,m

a

(

)

in H µω,*,m .

Theorem2.8. If fk is a bounded function on I = ( 0, ∞ ) then there exists n ∈ N 0 and

T ∈ ( H µω,*,m ) such that

T = ∑ Sµk [x −µ − 2 exp( λω(x))f k ] on H µω .

1

r=0

(

holds. Moreover, for every

continuous

k∈N0

Proof. The proof of the above result can be obtained from [1,pp. 46-47].

We denote H µω,* which is a union of space

and the interchange formula

h

(φ)}

Frechet space. So that in view of Theorem µ 2.7 and (2.9) it is clear that H ω is contained

N

then

−µ− 1 2

m,µ

it can be clearly shown that

From the above theorem if we take T ∈ H µω ' and φ ∈ H µω then T#φ belongs

' µ

{α

semi-norms

(T # φ ) ( x ) ∈ θ ω

( )

)

(2.9) is a linear space

and the topology generated by the family

This implies that

x −µ−

(

for k ∈ N0. Clearly H

µ ,* ω ,m

µ ω

( H ) endowed µ ,* ω,m

)

with

the

inductive

topology. We now characterize the element of H µω ' that belong to H µω ' .

( )

( )

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)

S. K. Upadhyay, J. Comp. & Math. Sci. Vol.2 (1), 145-152 (2011)

( )

151

Theorem2.9. Let T ∈ H µω ' . The following

φ→ h µ' (f # φ) is continuous linear mapping

conditions are equivalent: (i) T ∈ H µω '

µ µ ' from Qω into H ω . Finally, since h µ

( )

−µ− 1

µ

reduces to hµ on Qω , we deduce from

h µ' (T) ∈ θ Qµ

Theorem 2.1 that φ → f # φ is continuous

(ii)

x

(iii)

For every m ∈ N0 there exists r ∈ N0 and bounded continuous function fk on I such that

2

ω

r

T = ∑ S [exp(λω(x))f k ] k µ

k =0 λω (x)

and e every k. (iv)

(v)

(2.10)

f k is bounded on I for

For every m ∈ N0 there exists r ∈ N0 and bounded continuous functions fk on I, k ∈ N0 such that (2.10) holds λω(x) and e f k →∞ for every k.

For all m ∈ N0 there exists r ∈ N0 and bounded continuous functions fk on I such that (2.10) holds and eλω(x) fk is absolutely integrable on I for each k.

Proof. We omit the proof of the above results because it,s proof is entirely similar to that corresponding result of [1, pp. 48-50].

(

)

Theorem 2.10. Let f ∈ H µω,* ' . Then the µ

mapping φ→ f # φ continuous from H ω into itself.

Proof. By Theorem 2.5 for every φ∈Hµω we have

h µ' (f # φ ) = x

−µ−

1 2

h µ f .h µ φ

Using Theorem 2.1 and Theorem 2.9, we ' µ can show h µ (f # φ)∈Qω . Moreover,

µ

linear map of H ω into itself.

Theorem 2.11. Let f ∈ (H µω ) ' and g ∈ (H µω ) ' . T h en h µ' (f # g ) = x

−µ−

1 2

. h µ' f .h µ' g

Proof. From the definition of generalized Hankel transformation and Theorem (2.7), Theorem (2.10), we can write

h µ' (f # g), Φ = f # g, h µ (Φ ) = f , g # h µ (Φ ) = h µ' f , h µ' (g # h µ (Φ )) REFERENCES 1.

2.

3.

4.

5.

Betancor, J.J. and Mesha, L.R.: Hankel convolution on distributional spaces with exponential growth, Studia Mathematica, 121(1), 35-51(1996).

Beurling, A.: Quasi-analyticity and General lectures 4 and 5, AMS Stanford (1961).

Björck, G.: Linear partial differential operators and generalized distributions, Ark, Math, 6, 351-407 (1966).

Komotsu, H.: Ultradistribution, Istructure theorem and a characterization, J. Fac. Sci. Univ., Tokyo, 20, 25-105 (1973).

Pathak, R.S. and Pandey, A.B.: OnHankel transforms of ultradistributions, Applicable Analysis. 20, 245-262 (1985).

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)

152 6.

S. K. Upadhyay, J. Comp. & Math. Sci. Vol.2 (1), 145-152 (2011)

Roumieu, C.: Ultradistributions defines Sur Rn et Sur Certaines classes devarieties Differentiables J. d, Analyse Math. 10, 153-162 (1962-63).

7.

Zemanian, A.H.: Generalized integral transformations, Interscience shers, New York (1968).

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)

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